X Contents 6.8.3 Approximation:decouplingofrotationandvibrations 244 6.8.4 The kinetic energy operators of translation, rotation and vibrations . 245 6.8.5 Separationoftranslational,rotationalandvibrationalmotions 246 6.9 Non-boundstates 247 6.10Adiabatic,diabaticandnon-adiabaticapproaches 252 6.11Crossingofpotentialenergycurvesfordiatomics 255 6.11.1Thenon-crossingrule 255 6.11.2 Simulating the harpooning effect in the NaCl molecule 257 6.12Polyatomicmoleculesandconicalintersection 260 6.12.1Conicalintersection 262 6.12.2Berryphase 264 6.13Beyondtheadiabaticapproximation 268 6.13.1Muoncatalyzednuclearfusion 268 6.13.2“Russiandolls”–oramoleculewithinmolecule 270 7. MotionofNuclei 275 7.1 Rovibrational spectra – an example of accurate calculations: atom – di- atomicmolecule 278 7.1.1 CoordinatesystemandHamiltonian 279 7.1.2 Anisotropy of the potential V 280 7.1.3 Addingtheangularmomentainquantummechanics 281 7.1.4 ApplicationoftheRitzmethod 282 7.1.5 Calculationofrovibrationalspectra 283 7.2 Forcefields(FF) 284 7.3 LocalMolecularMechanics(MM) 290 7.3.1 Bondsthatcannotbreak 290 7.3.2 Bondsthatcanbreak 291 7.4 Globalmolecularmechanics 292 7.4.1 Multipleminimacatastrophe 292 7.4.2 Isittheglobalminimumwhichcounts? 293 7.5 Smallamplitudeharmonicmotion–normalmodes 294 7.5.1 Theoryofnormalmodes 295 7.5.2 Zero-vibrationenergy 303 7.6 MolecularDynamics(MD) 304 7.6.1 TheMDidea 304 7.6.2 WhatdoesMDofferus? 306 7.6.3 Whattoworryabout? 307 7.6.4 MD of non-equilibrium processes 308 7.6.5 Quantum-classicalMD 308 7.7 Simulatedannealing 309 7.8 LangevinDynamics 310 7.9 MonteCarloDynamics 311 7.10Car–Parrinellodynamics 314 7.11Cellularautomata 317 8. ElectronicMotionintheMeanField:AtomsandMolecules 324 8.1 Hartree–Fock method – a bird’s eye view 329 8.1.1 Spinorbitals 329 Contents XI 8.1.2 Variables 330 8.1.3 Slaterdeterminants 332 8.1.4 What is the Hartree–Fock method all about? 333 8.2 TheFockequationforoptimalspinorbitals 334 8.2.1 DiracandCoulombnotations 334 8.2.2 Energyfunctional 334 8.2.3 Thesearchfortheconditionalextremum 335 8.2.4 ASlaterdeterminantandaunitarytransformation 338 8.2.5 Invariance of the ˆ J and ˆ K operators 339 8.2.6 DiagonalizationoftheLagrangemultipliersmatrix 340 8.2.7 The Fock equation for optimal spinorbitals (General Hartree–Fock method–GHF) 341 8.2.8 The closed-shell systems and the Restricted Hartree–Fock (RHF) method 342 8.2.9 Iterative procedure for computing molecular orbitals: the Self- ConsistentFieldmethod 350 8.3 Total energy in the Hartree–Fock method 351 8.4 Computational technique: atomic orbitals as building blocks of the molecu- larwavefunction 354 8.4.1 Centringoftheatomicorbital 354 8.4.2 Slater-typeorbitals(STO) 355 8.4.3 Gaussian-typeorbitals(GTO) 357 8.4.4 LinearCombinationofAtomicOrbitals(LCAO)Method 360 8.4.5 BasissetsofAtomicOrbitals 363 8.4.6 The Hartree–Fock–Roothaan method (SCF LCAO MO) 364 8.4.7 PracticalproblemsintheSCFLCAOMOmethod 366 RESULTS OF THE HARTREE–FOCK METHOD 369 8.5 Backtofoundations 369 8.5.1 WhendoestheRHFmethodfail? 369 8.5.2 Fukutomeclasses 372 8.6 MendeleevPeriodicTableofChemicalElements 379 8.6.1 Similartothehydrogenatom–theorbitalmodelofatom 379 8.6.2 Yettherearedifferences 380 8.7 Thenatureofthechemicalbond 383 8.7.1 H + 2 intheMOpicture 384 8.7.2 Canweseeachemicalbond? 388 8.8 Excitation energy, ionization potential, and electron affinity (RHF approach) 389 8.8.1 Approximateenergiesofelectronicstates 389 8.8.2 Singletortripletexcitation? 391 8.8.3 Hund’srule 392 8.8.4 Ionization potential and electron affinity (Koopmans rule) 393 8.9 LocalizationofmolecularorbitalswithintheRHFmethod 396 8.9.1 Theexternallocalizationmethods 397 8.9.2 Theinternallocalizationmethods 398 8.9.3 Examplesoflocalization 400 8.9.4 Computationaltechnique 401 8.9.5 The σ, π, δ bonds 403 8.9.6 Electronpairdimensionsandthefoundationsofchemistry 404 8.9.7 Hybridization 407 XII Contents 8.10Aminimalmodelofamolecule 417 8.10.1ValenceShellElectronPairRepulsion(VSEPR) 419 9. ElectronicMotionintheMeanField:PeriodicSystems 428 9.1 Primitivelattice 431 9.2 Wavevector 433 9.3 Inverselattice 436 9.4 FirstBrillouinZone(FBZ) 438 9.5 PropertiesoftheFBZ 438 9.6 AfewwordsonBlochfunctions 439 9.6.1 Wavesin1D 439 9.6.2 Wavesin2D 442 9.7 Theinfinitecrystalasalimitofacyclicsystem 445 9.8 Atripleroleofthewavevector 448 9.9 Bandstructure 449 9.9.1 Born–vonKármánboundaryconditionin3D 449 9.9.2 CrystalorbitalsfromBlochfunctions(LCAOCOmethod) 450 9.9.3 SCFLCAOCOequations 452 9.9.4 Bandstructureandbandwidth 453 9.9.5 Fermi level and energy gap: insulators, semiconductors and metals . 454 9.10Solidstatequantumchemistry 460 9.10.1Whydosomebandsgoup? 460 9.10.2Whydosomebandsgodown? 462 9.10.3Whydosomebandsstayconstant? 462 9.10.4Howcanmorecomplexbehaviourbeexplained? 462 9.11 The Hartree–Fock method for crystals 468 9.11.1Secularequation 468 9.11.2IntegrationintheFBZ 471 9.11.3Fockmatrixelements 472 9.11.4Iterativeprocedure 474 9.11.5Totalenergy 474 9.12Long-rangeinteractionproblem 475 9.12.1Fockmatrixcorrections 476 9.12.2Totalenergycorrections 477 9.12.3MultipoleexpansionappliedtotheFockmatrix 479 9.12.4Multipoleexpansionappliedtothetotalenergy 483 9.13Backtotheexchangeterm 485 9.14Choiceofunitcell 488 9.14.1Fieldcompensationmethod 490 9.14.2Thesymmetryofsubsystemchoice 492 10.CorrelationoftheElectronicMotions 498 VARIATIONAL METHODS USING EXPLICITLY CORRELATED WAVE FUNC- TION 502 10.1 Correlationcuspcondition 503 10.2 TheHylleraasfunction 506 10.3 TheHylleraasCImethod 506 10.4 Theharmonicheliumatom 507 Contents XIII 10.5 James–CoolidgeandKołos–Wolniewiczfunctions 508 10.5.1 Neutrinomass 511 10.6 MethodofexponentiallycorrelatedGaussianfunctions 513 10.7 Coulombhole(“correlationhole”) 513 10.8 Exchangehole(“Fermihole”) 516 VARIATIONAL METHODS WITH SLATER DETERMINANTS 520 10.9 Valencebond(VB)method 520 10.9.1 Resonancetheory–hydrogenmolecule 520 10.9.2 Resonancetheory–polyatomiccase 523 10.10Configurationinteraction(CI)method 525 10.10.1Brillouintheorem 527 10.10.2ConvergenceoftheCIexpansion 527 10.10.3 Example of H 2 O 528 10.10.4Whichexcitationsaremostimportant? 529 10.10.5Naturalorbitals(NO) 531 10.10.6Sizeconsistency 532 10.11DirectCImethod 533 10.12MultireferenceCImethod 533 10.13MulticonfigurationalSelf-ConsistentFieldmethod(MCSCF) 535 10.13.1ClassicalMCSCFapproach 535 10.13.2UnitaryMCSCFmethod 536 10.13.3Completeactivespacemethod(CASSCF) 538 NON-VARIATIONAL METHODS WITH SLATER DETERMINANTS 539 10.14 Coupled cluster (CC) method 539 10.14.1Waveandclusteroperators 540 10.14.2RelationshipbetweenCIandCCmethods 542 10.14.3SolutionoftheCCequations 543 10.14.4 Example: CC with double excitations 545 10.14.5SizeconsistencyoftheCCmethod 547 10.15Equation-of-motionmethod(EOM-CC) 548 10.15.1Similaritytransformation 548 10.15.2DerivationoftheEOM-CCequations 549 10.16Manybodyperturbationtheory(MBPT) 551 10.16.1UnperturbedHamiltonian 551 10.16.2Perturbationtheory–slightlydifferentapproach 552 10.16.3 Reduced resolvent or the “almost” inverse of (E (0) 0 − ˆ H (0) ) 553 10.16.4MBPTmachinery 555 10.16.5Brillouin–Wignerperturbationtheory 556 10.16.6Rayleigh–Schrödingerperturbationtheory 557 10.17Møller–PlessetversionofRayleigh–Schrödingerperturbationtheory 558 10.17.1ExpressionforMP2energy 558 10.17.2ConvergenceoftheMøller–Plessetperturbationseries 559 10.17.3 Special status of double excitations 560 11.ElectronicMotion:DensityFunctionalTheory(DFT) 567 11.1 Electronicdensity–thesuperstar 569 11.2 Baderanalysis 571 11.2.1 Overall shape of ρ 571 XIV Contents 11.2.2 Criticalpoints 571 11.2.3 Laplacianoftheelectronicdensityasa“magnifyingglass” 575 11.3 TwoimportantHohenberg–Kohntheorems 579 11.3.1 Equivalence of the electronic wave function and electron density . 579 11.3.2 Existence of an energy functional minimized by ρ 0 580 11.4 TheKohn–Shamequations 584 11.4.1 TheKohn–Shamsystemofnon-interactingelectrons 584 11.4.2 Totalenergyexpression 585 11.4.3 DerivationoftheKohn–Shamequations 586 11.5 What to take as the DFT exchange–correlation energy E xc ? 590 11.5.1 Localdensityapproximation(LDA) 590 11.5.2 Non-localapproximations(NLDA) 591 11.5.3 The approximate character of the DFT vs apparent rigour of ab initio computations 592 11.6 Onthephysicaljustificationfortheexchangecorrelationenergy 592 11.6.1 Theelectronpairdistributionfunction 592 11.6.2 Thequasi-staticconnectionoftwoimportantsystems 594 11.6.3 Exchange–correlation energy vs  aver 596 11.6.4 Electronholes 597 11.6.5 Physicalboundaryconditionsforholes 598 11.6.6 Exchangeandcorrelationholes 599 11.6.7 PhysicalgroundsfortheDFTapproximations 601 11.7 ReflectionsontheDFTsuccess 602 12.TheMoleculeinanElectricorMagneticField 615 12.1 Hellmann–Feynmantheorem 618 ELECTRIC PHENOMENA 620 12.2 The molecule immobilized in an electric field 620 12.2.1 Theelectricfieldasaperturbation 621 12.2.2 Thehomogeneouselectricfield 627 12.2.3 The inhomogeneous electric field: multipole polarizabilities and hyperpolarizabilities . 632 12.3 Howtocalculatethedipolemoment 633 12.3.1 Hartree–Fock approximation 633 12.3.2 Atomicandbonddipoles 634 12.3.3 WithintheZDOapproximation 635 12.4 How to calculate the dipole polarizability 635 12.4.1 SumOverStatesMethod 635 12.4.2 Finitefieldmethod 639 12.4.3 Whatisgoingonathigherelectricfields 644 12.5 A molecule in an oscillating electric field 645 MAGNETIC PHENOMENA 647 12.6 Magneticdipolemomentsofelementaryparticles 648 12.6.1 Electron 648 12.6.2 Nucleus 649 12.6.3 Dipolemomentinthefield 650 12.7 Transitions between the nuclear spin quantum states – NMR technique . . 652 12.8 Hamiltonianofthesystemintheelectromagneticfield 653 Contents XV 12.8.1 Choiceofthevectorandscalarpotentials 654 12.8.2 RefinementoftheHamiltonian 654 12.9 EffectiveNMRHamiltonian 658 12.9.1 Signalaveraging 658 12.9.2 EmpiricalHamiltonian 659 12.9.3 Nuclearspinenergylevels 664 12.10TheRamseytheoryoftheNMRchemicalshift 666 12.10.1Shieldingconstants 667 12.10.2Diamagneticandparamagneticcontributions 668 12.11TheRamseytheoryofNMRspin–spincouplingconstants 668 12.11.1Diamagneticcontributions 669 12.11.2Paramagneticcontributions 670 12.11.3Couplingconstants 671 12.11.4TheFermicontactcouplingmechanism 672 12.12Gaugeinvariantatomicorbitals(GIAO) 673 12.12.1Londonorbitals 673 12.12.2Integralsareinvariant 674 13.IntermolecularInteractions 681 THEORY OF INTERMOLECULAR INTERACTIONS 684 13.1 Interactionenergyconcept 684 13.1.1 Naturaldivisionanditsgradation 684 13.1.2 Whatismostnatural? 685 13.2 Bindingenergy 687 13.3 Dissociationenergy 687 13.4 Dissociationbarrier 687 13.5 Supermolecular approach 689 13.5.1 Accuracyshouldbethesame 689 13.5.2 Basissetsuperpositionerror(BSSE) 690 13.5.3 Good and bad news about the supermolecular method 691 13.6 Perturbationalapproach 692 13.6.1 Intermoleculardistance–whatdoesitmean? 692 13.6.2 Polarizationapproximation(twomolecules) 692 13.6.3 Intermolecularinteractions:physicalinterpretation 696 13.6.4 Electrostatic energy in the multipole representation and the pene- trationenergy 700 13.6.5 Inductionenergyinthemultipolerepresentation 703 13.6.6 Dispersionenergyinthemultipolerepresentation 704 13.7 Symmetryadaptedperturbationtheories(SAPT) 710 13.7.1 Polarization approximation is illegal 710 13.7.2 Constructingasymmetryadaptedfunction 711 13.7.3 The perturbation is always large in polarization approximation . . 712 13.7.4 Iterative scheme of the symmetry adapted perturbation theory . . 713 13.7.5 Symmetryforcing 716 13.7.6 A link to the variational method – the Heitler–London interaction energy 720 13.7.7 When we do not have at our disposal the ideal ψ A0 and ψ B0 . . 720 13.8 Convergenceproblems 721 XVI Contents 13.9 Non-additivityofintermolecularinteractions 726 13.9.1 Many-bodyexpansionofinteractionenergy 727 13.9.2 Additivityoftheelectrostaticinteraction 730 13.9.3 Exchangenon-additivity 731 13.9.4 Inductionenergynon-additivity 735 13.9.5 Additivityofthesecond-orderdispersionenergy 740 13.9.6 Non-additivityofthethird-orderdispersioninteraction 741 ENGINEERING OF INTERMOLECULAR INTERACTIONS 741 13.10Noblegasinteraction 741 13.11VanderWaalssurfaceandradii 742 13.11.1PaulihardnessofthevanderWaalssurface 743 13.11.2Quantumchemistryofconfinedspace–thenanovessels 743 13.12Synthonsandsupramolecularchemistry 744 13.12.1Boundornotbound 745 13.12.2 Distinguished role of the electrostatic interaction and the valence repulsion 746 13.12.3Hydrogenbond 746 13.12.4Coordinationinteraction 747 13.12.5Hydrophobiceffect 748 13.12.6Molecularrecognition–synthons 750 13.12.7“Key-lock”,templateand“hand-glove”synthoninteractions 751 14.IntermolecularMotionofElectronsandNuclei:ChemicalReactions 762 14.1 Hypersurfaceofthepotentialenergyfornuclearmotion 766 14.1.1 Potentialenergyminimaandsaddlepoints 767 14.1.2 Distinguishedreactioncoordinate(DRC) 768 14.1.3 Steepestdescentpath(SDP) 769 14.1.4 Ourgoal 769 14.1.5 Chemicalreactiondynamics(apioneers’approach) 770 14.2 Accuratesolutionsforthereactionhypersurface(threeatoms) 775 14.2.1 CoordinatesystemandHamiltonian 775 14.2.2 SolutiontotheSchrödingerequation 778 14.2.3 Berryphase 780 14.3 Intrinsicreactioncoordinate(IRC)orstatics 781 14.4 ReactionpathHamiltonianmethod 783 14.4.1 EnergyclosetoIRC 783 14.4.2 Vibrationallyadiabaticapproximation 785 14.4.3 Vibrationallynon-adiabaticmodel 790 14.4.4 Application of the reaction path Hamiltonian method to the reac- tion H 2 +OH →H 2 O +H 792 14.5 Acceptor–donor(AD)theoryofchemicalreactions 798 14.5.1 Mapsofthemolecularelectrostaticpotential 798 14.5.2 Wheredoesthebarriercomefrom? 803 14.5.3 MO,ADandVBformalisms 803 14.5.4 Reactionstages 806 14.5.5 Contributionsofthestructuresasreactionproceeds 811 14.5.6 Nucleophilic attack H − + ETHYLENE → ETHYLENE + H − . . 816 14.5.7 Electrophilic attack H + + H 2 → H 2 + H + 818 Contents XVII 14.5.8 Nucleophilic attack on the polarized chemical bond in the VB pic- ture 818 14.5.9 Whatisgoingoninthechemist’sflask? 821 14.5.10Roleofsymmetry 822 14.5.11Barriermeansacostofopeningtheclosed-shells 826 14.6 Barrierfortheelectron-transferreaction 828 14.6.1 Diabaticandadiabaticpotential 828 14.6.2 Marcustheory 830 15.InformationProcessing–theMissionofChemistry 848 15.1 Complexsystems 852 15.2 Self-organizingcomplexsystems 853 15.3 Cooperative interactions 854 15.4 Sensitivityanalysis 855 15.5 Combinatorialchemistry–molecularlibraries 855 15.6 Non-linearity 857 15.7 Attractors 858 15.8 Limitcycles 859 15.9 Bifurcationsandchaos 860 15.10Catastrophes 862 15.11Collectivephenomena 863 15.11.1Scalesymmetry(renormalization) 863 15.11.2Fractals 865 15.12Chemicalfeedback–non-linearchemicaldynamics 866 15.12.1Brusselator–dissipativestructures 868 15.12.2Hypercycles 873 15.13Functionsandtheirspace-timeorganization 875 15.14Themeasureofinformation 875 15.15Themissionofchemistry 877 15.16Molecularcomputersbasedonsynthoninteractions 878 APPENDICES 887 A. AREMAINDER:MATRICESANDDETERMINANTS 889 1.Matrices 889 2.Determinants 892 B. AFEWWORDSONSPACES,VECTORSANDFUNCTIONS 895 1.Vectorspace 895 2.Euclideanspace 896 3.Unitaryspace 897 4.Hilbertspace 898 5.Eigenvalueequation 900 C. GROUPTHEORYINSPECTROSCOPY 903 1.Group 903 2.Representations 913 XVIII Contents 3.Grouptheoryandquantummechanics 924 4.Integralsimportantinspectroscopy 929 D. ATWO-STATEMODEL 948 E. DIRACDELTAFUNCTION 951 1. Approximations to δ(x) 951 2. Properties of δ(x) 953 3.AnapplicationoftheDiracdeltafunction 953 F. TRANSLATION vs MOMENTUM and ROTATION vs ANGULAR MOMENTUM 955 1. The form of the ˆ U operator 955 2.TheHamiltoniancommuteswiththetotalmomentumoperator 957 3. The Hamiltonian, ˆ J 2 and ˆ J z docommute 958 4.Rotationandtranslationoperatorsdonotcommute 960 5.Conclusion 960 G. VECTORANDSCALARPOTENTIALS 962 H. OPTIMALWAVEFUNCTIONFORAHYDROGEN-LIKEATOM 969 I. SPACE-ANDBODY-FIXEDCOORDINATESYSTEMS 971 J. ORTHOGONALIZATION 977 1.Schmidtorthogonalization 977 2.Löwdinsymmetricorthogonalization 978 K. DIAGONALIZATIONOFAMATRIX 982 L. SECULAR EQUATION (H −εS)c =0 984 M.SLATER–CONDONRULES 986 N. LAGRANGEMULTIPLIERSMETHOD 997 O. PENALTYFUNCTIONMETHOD 1001 P. MOLECULARINTEGRALSWITHGAUSSIANTYPEORBITALS1s 1004 Q. SINGLETANDTRIPLETSTATESFORTWOELECTRONS 1006 Contents XIX R. THE HYDROGEN MOLECULAR ION IN THE SIMPLEST ATOMIC BASIS SET 1009 S. POPULATIONANALYSIS 1015 T. THEDIPOLEMOMENTOFALONEELECTRONPAIR 1020 U. SECONDQUANTIZATION 1023 V. THE HYDROGEN ATOM IN THE ELECTRIC FIELD – VARIATIONAL AP- PROACH 1029 W.NMRSHIELDINGANDCOUPLINGCONSTANTS–DERIVATION 1032 1.Shieldingconstants 1032 2.Couplingconstants 1035 X. MULTIPOLEEXPANSION 1038 Y. PAULIDEFORMATION 1050 Z. ACCEPTOR–DONOR STRUCTURE CONTRIBUTIONS IN THE MO CON- FIGURATION 1058 NameIndex 1065 SubjectIndex 1077 . 742 13.11.1PaulihardnessofthevanderWaalssurface 743 13.11.2Quantumchemistryofconfinedspace–thenanovessels 743 13.12Synthonsandsupramolecularchemistry 744 13.12.1Boundornotbound 745 13.12.2 Distinguished role of the. 270 7. MotionofNuclei 275 7.1 Rovibrational spectra – an example of accurate calculations: atom – di- atomicmolecule 278 7.1.1 CoordinatesystemandHamiltonian 279 7.1.2 Anisotropy of the potential. Anisotropy of the potential V 280 7.1.3 Addingtheangularmomentainquantummechanics 281 7.1.4 ApplicationoftheRitzmethod 282 7.1.5 Calculationofrovibrationalspectra 283 7.2 Forcefields(FF) 284 7.3 LocalMolecularMechanics(MM)